Membership function for a trapezoidal fuzzy number. Membership of 22

Figure 3.4

is 0.34.

We may derive a formula for a normal (bell-shaped) fuzzy number by using the

dispersion d=2 from its central value b to the points where the membership is 0.5.

[For the approximating triangular fuzzy number, (b À d) corresponds to the point

x ¼ a where the membership ¬rst begins to rise from zero, b to the central value,

and (b þ d) to the point c where the membership ¬rst reaches zero after that.] A

normal fuzzy number with dispersion d and central value b has formula (3.21) for

its membership function:

2 !

xÀb

(3:21)

exp ln(0:5)

d=2

The support of a fuzzy number is the interval between the point where the

membership ¬rst begins to increase from zero and the point at which the member-

ship last returns to zero. Thus in the fuzzy numbers above, the support is the interval

from c to a, except for normal fuzzy numbers whose support is in¬nite.

If a ¼ b ¼ c and d ¼ 0, the fuzzy number has grade of membership 1 only at

x ¼ b, and is 0 everywhere else. Such a fuzzy number is called a singleton. A single-

ton is the precise fuzzy counterpart of an ordinary scalar number, such as (say) 35 or

3.1416. The graph of the membership function of a singleton fuzzy number is not very

exciting; it has a single spike from membership zero to membership one (with zero

width) at one point, and is zero everywhere else. Note that all the symmetric fuzzy

numbers pass through the points where m(x) ¼ 0.5 at x ¼ b þ/2 d/2. Convention-

ally, membership functions for fuzzy numbers will be normalized, which means

that their membership function takes on the value one for some x.

3.3.3 Linguistic Variables and Membership Functions

As mentioned in 3.3.1, some discrete fuzzy sets describe numeric quantities.

Numeric discrete fuzzy sets have been formalized as linguistic variables, which

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43

3.4 FUZZY RELATIONS

Membership functions of a linguistic variable speed.

Figure 3.5

consist of the name of the discrete fuzzy set; the names of its members, known as

linguistic values; and for each linguistic value, a membership function like those

for fuzzy numbers. The universe on which the linguistic variable is de¬ned is

assumed to be the entire real line, although in speci¬c applications a smaller uni-

verse might be needed, such as all non-negative real numbers. (The original de¬-

nition of a linguistic variable (Zadeh, 1974) was somewhat broader, but our

de¬nition will suf¬ce.)

An example of a linguistic variable might be Speed, whose members are Slow,

Medium, and Fast, and whose membership functions are shown in Figure 3.5.

Linguistic variables have attracted much attention in the fuzzy literature, partly

because of their great importance in fuzzy control. The reader is cautioned that in

the fuzzy literature, the terminology has not been consistent; linguistic values

have also been called linguistic labels and linguistic terms. In addition, while mem-

bership functions are of course fuzzy sets with an in¬nite number of members, we

have seen that they are only one kind of fuzzy set. However, because of the great

interest in fuzzy control, the term “fuzzy set” in the literature often is assumed to

be synonymous with membership function. Usually, the context will make clear

what is meant.

3.4 FUZZY RELATIONS

3.4.1 Matrices of Truth Values

We will have several situations in which we deal with arrays of truth values. For

example, if we have digitized a membership function by sampling it at discrete

values of its numeric argument, we have created a vector of truth values. In the fol-

lowing section on fuzzy relations, we will deal with matrices of truth values. We will

call these fuzzy matrices. We now de¬ne an important operation we can employ on

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44 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: I

such arrays. Roughly speaking, the operation of addition in ordinary matrices is

analogous to the fuzzy logical OR, and the operation of multiplication on ordinary

matrices is analogous to the fuzzy logical AND.

The most important operation on matrices of truth values is called composition,

and is analogous to matrix multiplication. Suppose I have two fuzzy matrices A and

B. To compose these matrices, they must meet the same size compatibility restric-

tions as in ordinary matrix multiplication. Let A have l rows and m columns, and let

B have m rows and n columns. They are compatible, since the number of columns of

A equals the number of rows of B. The composition of A and B is denoted A W B, and

will produce a matrix C having l rows and n columns.

In ordinary matrix multiplication, we would obtain cij by summing the product

aik ‚ bkj over k. We could write this as

ci, j ¼ ai,1 Á b1, j þ ai,2 Á b2, j þ Á Á Á þ ai,n Á bn, j

In fuzzy matrix composition, we obtain cij by repeatedly ORing (aik AND bkj) over

all values of k, using min “ max logic by default. This procedure gives:

cij ¼ (ai,1 AND b1, j ) OR (ai,2 AND b2, j ) OR Á Á Á OR (ai,n AND bn, j ) (3:22)

The operation of composing matrices A and B is written C ¼ A W B. For example,

suppose we have these two fuzzy matrices to be composed:

0:2 0:4 0:6

A¼

0:3 0:6 0:9

2 3

(3:23)

0:5 1

6 7

B ¼ 4 0:7 0:5 5

1 0

We ¬rst compute c1,1, and apply (3.22) to the ¬rst row of A and the ¬rst column

of B. The minimum of 0.2 and 0.5 is 0.2; min(0.4, 0.7) is 0.4; and min(0.6, 1) is

0.6; and the maximum of these minima is 0.6. Continuing this procedure, we

obtain C as:

0:6 0:4

C¼ (3:24)

0:9 0:5

3.4.2 Relations Between Sets

We have two universal sets X and Y. By X ‚ Y we mean the set of all ordered pairs

(x, y) for x in X and y in Y. A fuzzy relation R on X ‚ Y is a fuzzy subset of X ‚ Y;

that is, for each (x, y) pair we have a number ranging from 0 to 1, a measure of the

relationship between x and y.

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45

3.4 FUZZY RELATIONS

As a simple example of a fuzzy relation let X ¼ {John, Jim, Bill} and